Uniformly most powerful test

Remarks

Here $E_\theta[\varphi(X)]$ is the power function of $\varphi$. Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\theta, 1)$, $H_0 \colon \theta \le 0$, $H_1 \colon \theta > 0$. The test that rejects when $\sqrt{n}\, \bar X > z_{1 - \alpha}$, the $1 - \alpha$ quantile of $\mathcal{N}(0, 1)$, is UMP at level $\alpha$. For each $\theta_1 > 0$ it is the most powerful test of $\theta = 0$ against $\theta = \theta_1$, by the Neyman–Pearson lemma, and its power function is increasing. Non-example: for $H_0 \colon \theta = 0$ against $H_1 \colon \theta \ne 0$ in the same model, no test satisfies clause 2. The one-sided test above is most powerful against each $\theta_1 > 0$, but at each $\theta_1 < 0$ the mirror one-sided test has strictly more power.